SPS SPS SM 2021 February — Question 7 15 marks

Exam BoardSPS
ModuleSPS SM (SPS SM)
Year2021
SessionFebruary
Marks15
TopicNormal Distribution
TypeMultiple probability calculations only
DifficultyStandard +0.3 This is a standard A-level statistics question covering normal distribution calculations and hypothesis testing. Part (a) is routine z-score calculation, part (b) is textbook hypothesis testing with known variance, parts (c)(i-iii) involve straightforward conditional probability, and part (d) requires finding the median of a truncated normal distribution. While part (d) involves slightly more sophisticated reasoning about truncated distributions, all techniques are standard S2/S3 level with no novel problem-solving required. The multi-part structure and 15 total marks indicate moderate length but below-average conceptual difficulty.
Spec2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation2.05e Hypothesis test for normal mean: known variance
A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.
  1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. [1]
  2. Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes. Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients' complaint. [4]
  3. The health centre also claims that the time a dentist spends with a patient during a routine appointment, \(T\) minutes, can be modelled by the normal distribution where \(T \sim N(5, 3.5^2)\) Using this model,
    1. find the probability that a routine appointment with the dentist takes less than 2 minutes [1]
    2. find \(P(T < 2 | T > 0)\) [3]
    3. hence explain why this normal distribution may not be a good model for \(T\). [1]
  4. The dentist believes that she cannot complete a routine appointment in less than 2 minutes. She suggests that the health centre should use a refined model only including values of \(T > 2\) Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place. [5]
A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.

\begin{enumerate}[label=(\alph*)]
\item Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes.
[1]

\item Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes.
The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes.
Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients' complaint.
[4]

\item The health centre also claims that the time a dentist spends with a patient during a routine appointment, $T$ minutes, can be modelled by the normal distribution where $T \sim N(5, 3.5^2)$
Using this model,
\begin{enumerate}[label=(\roman*)]
\item find the probability that a routine appointment with the dentist takes less than 2 minutes
[1]

\item find $P(T < 2 | T > 0)$
[3]

\item hence explain why this normal distribution may not be a good model for $T$.
[1]
\end{enumerate}

\item The dentist believes that she cannot complete a routine appointment in less than 2 minutes.
She suggests that the health centre should use a refined model only including values of $T > 2$
Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place.
[5]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM 2021 Q7 [15]}}
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